Estimating the Likelihood of Purchase
At the heart of our method for predicting consumer behavior is what we call the likelihood function. The function estimates the likelihood (Li) that a customer or household (i) will purchase a given product at a given time:
Ri is the number of interpurchase times for customer or household i
if the ri th interpurchase time extends
beyond the observation window
if product j is bought by customer or household i at time t; the probability that ijt =1 is Pij (t)
otherwise; the probability that ijt = 0 is (1—Pij (t))
and ƒi (•) and Si (•) denote the density and survivor functions, respectively. The term involving Si (t) accounts for right censoring of the data, because the end of the data collection period usually does not coincide with a purchase for all households. Consequently, this term does not depend on Pij (t). Standard maximum likelihood methods can be used to estimate the model parameters. We have implemented the model-estimation procedure in a Gauss program.